As part of a more general conjecture by A. Katok and R.
Spatzier, the following statement was expected to hold: if a smooth
$\mathbb Z^r$-action $\alpha$ on a torus contains one Anosov element
and has no rank-1 factor, then it must be smoothly conjugate to its
linearization $\alpha_0$, which is an action by toral automorphisms.
D. Fisher, B. Kalinin and R. Spatzier showed this holds under the
assumption that $\alpha$ has at least one Anosov element in every Weyl
chamber of the linearization action. We will verify that this
assumption is redundant, hence fully establish the statement above.
This is a joint work with Federico Rodriguez Hertz.